Question 1183504: A farming cooperative in the KwaMashu buys wheat seeds for its farmer members from seed merchants. A particular seed merchant claims that their wheat seeds have at least an 80% germination rate. Before the farming cooperative will buy from this seed merchant, they want to verify this claim. A random sample of 320 wheat seeds supplied by this seed merchant was tested, and it was found that only 230 seeds germinated. Is there sufficient statistical evidence at the 3% significance level to justify the purchase of wheat seeds from this seed merchant? Use the p-value approach to conduct a hypothesis test for a single proportion, and report the findings to the KwaMashu farming cooperative
Found 2 solutions by Boreal, Theo:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! 230/320=p hat, the best estimate=0.71875
Ho:p=0.80
Ha: p NE 0.80
alpha=0.03
test statistic is a z, reject this two way test if |z| > 2.17
z=(0.71875-0.80)/sqrt(0.8*0.2/320)
=-0.08125/0.0224
=-3.63
It does not justify purchase of the wheat seeds.
reject Ho with p-value 0.0003 (double the area from -oo to -3.63 since two way test)
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the claimed ratio is .8
the sample ratio is 230/320 = .71875
standard error = square root of (p*q/n)
p is the claimed population ratio.
q is equal to 1 - p.
n is the sample size.
standard error = square root of (.8 * .2 / 320) = .02236 rounded to 5 decimal places.
the z-score is equal to (.71875 - .8) / .02236 = -3.6337 rounded to 4 decimal places.
the area to the left of that z-score is equal to .0001397 rounded to 7 decimal places.
the critical alpha is equal to .015.
that is the area to the left of the critical z-score.
since the area to the left of the test z-score is less than the critical area, the claim of .8 is rejected.
that should be your answer.
the two tailed critical z-score at 3% confidence level is plus or minus 2.17.
since the test z-score is -3.6337, that is further out than the critical z-score, therefore the .8 claim is rejected.
in general, if you compare your z-score to the critical z-scores, the null hypothesis is rejected if the absolute value of the test z-score is greater than the absolute value of the critical z-score.
alternatively, if you compare the test alpha to the critical alpha, the null hypothesis is rejected if the test alpha is less than the critical alpha.
in both cases, the test z-score / alpha is outside the limits of the critical z-score / alpha.
here's a reference that you might find helpful.
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Statistics_Using_Technology_(Kozak)/07%3A_One-Sample_Inference/7.02%3A_One-Sample_Proportion_Test
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